## Left Regular Representation (Pt. II)

**Point of post: **This post a is a continuation of this one.

*Left Multiplication Representation*

With the above inner product we can now successfully describe the canonical -representation of into endowed with the canonical inner product. Namely:

**Theorem: ***Let be a finite group and be the group algebra on . Then, if is given the canonical inner product then the map given by is a -representation.*

**Proof: **First of all it’s worth showing that the map really does define a linear transformation. But, this is fairly trivial to check since for any and

It thus remains to show that is a -representation of . To see that is additive we merely note that for any we have that

from where it follows from the arbitrariness of that . Similarly one has that

and thus by the arbitrariness of it follows that . Noticing that evidently for any one has that it’s clear that . Thus, it remains show that that . To do this we recall that is the unique endomorphism on such that

for all . Thus, it suffices to show that satisfies this property. Indeed, let and note that

from where by the arbitrariness of we are able to conclude, by the aforementioned characterization of the adjoint, that . The theorem then follows.

*Left Regular Representation*

We have just seen that the map given by is a -representation on . But, by previous theorem we know that this induces a representation given by . At first this may seem like quite an intractable definition since, in practice, means nothing. What’s interesting is that acts on elements of in a particularly simple way. In particular, . Indeed:

This representation is called the *left regular representation *of on .

Now, while this is an enlightening way to look at the left regular representation there is an admittedly easier method to prove that it’s a representation. Namely, it’s clear that if is the left regular representation then is a homomorphism and thus it would suffice to check that is unitary for each . Note though that from where it clearly follows that takes the orthonormal basis to itself, and thus (by a common characterization of unitarity) each is unitary.

We shall see that the left regular representation will play a decisive role in the theory to come.

**References:**

1.Simon, Barry. *Representations of Finite and Compact Groups*. Providence, RI: American Mathematical Society, 1996. Print.

2. Serre, Jean Pierre. *Linear Representations of Finite Groups*. New York: Springer-Verlag, 1977. Print

[…] Let . Define by where denotes the usual left representation of into and is the usual inner product on the group algebra. Now, from previous theorem we know […]

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